A factorial of a positive number n is defined to be the product of all positive whole numbers less than and equal to n. E.g. n=5 5!=5×4×3×2×1. Note: We define the factorial 0!=1.
Calculate 7!.
From the definition we have 7!=7×6×5×4×3×2×1. Hence 7!=5040.
Simplify 9!6!.
Firstly we can expand the factorials using the definition 9!6!=9×8×7×6×5×4×3×2×16×5×4×3×2×1. This can then be simplified by canceling terms on the top and bottom to give 9!6!=9×8×7. Therefore, 9!6!=504.
For more information see simplifying fractions.
Simplify 16!13!⋅8!
As above start by expanding the factorials 16!13!⋅8!=16×15×⋯×3×2×1(13×12×11×⋯×3×2×1)(8×7×6×5×4×3×2×1). Then cancel out any terms that appear on the top and the bottom of the fraction 16!13!⋅8!=16×15×148×7×6×5×4×3×2. This can be canceled further if we factorize the numerator 16!13!⋅8!=8×2×5×3×2×78×7×6×5×4×3×2. Then finally we get 16!13!⋅8!=14×3,=112.
Simplify (n−1)!(n+1)!.
(n−1)!(n+1)!=(n−1)×(n−2)×⋯×2×1(n+1)×n×(n−1)×⋯×2×1,=1(n+1)×n,=1n2+n.
The binomial expansion uses factorials to calculate the coefficients.
Try our Numbas test on factorials.